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Derivative with a New Parameter

Theory, Methods and Applications

  • 1st Edition - September 10, 2015
  • Latest edition
  • Author: Abdon Atangana
  • Language: English

Derivative with a New Parameter: Theory, Methods and Applications discusses the first application of the local derivative that was done by Newton for general physics, and later fo… Read more

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Description

Derivative with a New Parameter: Theory, Methods and Applications discusses the first application of the local derivative that was done by Newton for general physics, and later for other areas of the sciences.

The book starts off by giving a history of derivatives, from Newton to Caputo. It then goes on to introduce the new parameters for the local derivative, including its definition and properties. Additional topics define beta-Laplace transforms, beta-Sumudu transforms, and beta-Fourier transforms, including their properties, and then go on to describe the method for partial differential with the beta derivatives.

Subsequent sections give examples on how local derivatives with a new parameter can be used to model different applications, such as groundwater flow and different diseases. The book gives an introduction to the newly-established local derivative with new parameters, along with their integral transforms and applications, also including great examples on how it can be used in epidemiology and groundwater studies.

Key features

  • Introduce the new parameters for the local derivative, including its definition and properties
  • Provides examples on how local derivatives with a new parameter can be used to model different applications, such as groundwater flow and different diseases
  • Includes definitions of beta-Laplace transforms, beta-Sumudu transforms, and beta-Fourier transforms, their properties, and methods for partial differential using beta derivatives
  • Explains how the new parameter can be used in multiple methods

Readership

Scientists and engineers in the fields of mathematics, physics, chemistry and engineering

Table of contents

  • Dedication
  • Preface
  • Acknowledgments
  • Chapter 1: History of derivatives from Newton to Caputo
    • Abstract
    • 1.1 Introduction
    • 1.2 Definition of local and fractional derivative
    • 1.3 Definitions and properties of their anti-derivatives
    • 1.4 Limitations and strength of local and fractional derivatives
    • 1.5 Classification of fractional derivatives
  • Chapter 2: Local derivative with new parameter
    • Abstract
    • 2.1 Motivation
    • 2.2 Definition and anti-derivative
    • 2.3 Properties of local derivative with new parameter
    • 2.4 Definition of partial derivative with new parameter
    • 2.5 Properties of partial beta-derivatives
  • Chapter 3: Novel integrals transform
    • Abstract
    • 3.1 Definition of some integral transform operators
    • 3.2 Definition and properties of the beta-Laplace transform
    • 3.3 Definition and properties of the beta-Sumudu transform
    • 3.4 Definition and properties of beta-Fourier transform
  • Chapter 4: Method for partial differential equations with beta-derivative
    • Abstract
    • 4.1 Introduction
    • 4.2 Homotopy decomposition method
    • 4.3 Variational iteration method
    • 4.4 Sumudu decomposition method
    • 4.5 Laplace decomposition method
    • 4.6 Extension of match asymptotic method to fractional boundary layers problems
    • 4.7 Numerical method
    • 4.8 Generalized stationarity with a new parameter
  • Chapter 5: Applications of local derivative with new parameter
    • Abstract
    • 5.1 Introduction
    • 5.2 Model of groundwater flow within the confined aquifer
    • 5.3 Steady-state solutions of the flow in a confined and unconfined aquifer
    • 5.4 Model of groundwater flow equation within a leaky aquifer
    • 5.5 Model of Lassa fever or Lassa hemorrhagic fever
    • 5.6 Model of Ebola hemorrhagic fever
  • Bibliography

Review quotes

"To define a new integral transform, in addition to the definition one must provide an inversion theorem, existence and uniqueness theorems."—Zentralblatt MATH

Product details

  • Edition: 1
  • Latest edition
  • Published: September 24, 2015
  • Language: English

About the author

AA

Abdon Atangana

Dr. Abdon Atangana is Academic Head of Department and Professor of Applied Mathematics at the University of the Free State, Bloemfontein, Republic of South Africa. He obtained his honours and master’s degrees from the Department of Applied Mathematics at the UFS with distinction. He obtained his PhD in applied mathematics from the Institute for Groundwater Studies. He was included in the 2019 (Maths), 2020 (Cross-field) and the 2021 (Maths) Clarivate Web of Science lists of the World's top 1% scientists, and he was awarded The World Academy of Sciences (TWAS) inaugural Mohammed A. Hamdan award for contributions to science in developing countries. In 2018 Dr. Atangana was elected as a member of the African Academy of Sciences and in 2021 a member of The World Academy of Sciences. He also ranked number one in the world in mathematics, number 186 in the world in all fields, and number one in Africa in all fields, according to the Stanford University list of top 2% scientists in the world. He was one of the first recipients of the Obada Award in 2018. Dr. Atangana published a paper that was ranked by Clarivate in 2017 as the most cited mathematics paper in the world. Dr. Atangana serves as an editor for 20 international journals, lead guest editor for 10 journals, and is also a reviewer of more than 200 international accredited journals. His research interests include methods and applications of partial and ordinary differential equations, fractional differential equations, perturbation methods, asymptotic methods, iterative methods, and groundwater modelling. Dr. Atangana is a pioneer in research on fractional calculus with non-local and non-singular kernels popular in applied mathematics today. He is the author of numerous books, including Integral Transforms and Engineering: Theory, Methods, and Applications, Taylor and Francis/CRC Press; Numerical Methods for Fractal-Fractional Differential Equations and Engineering: Simulations and Modeling, Taylor and Francis/CRC Press; Numerical Methods for Fractional Differentiation, Springer; Fractional Stochastic Differential Equations, Springer; Fractional Order Analysis, Wiley; Applications of Fractional Calculus to Modeling in Dynamics and Chaos, Taylor and Francis/CRC Press; Fractional Operators with Constant and Variable Order with Application to Geo-hydrology, Elsevier/Academic Press; Derivative with a New Parameter: Theory, Methods, and Applications, Elsevier/Academic Press; and New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications, Elsevier/Academic Press; among others.
Affiliations and expertise
Academic Head, Department and Professor of Applied Mathematics, University of the Free State, Bloemfontein, South Africa

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